Noncomutative Discrete Geometric Analysis

离散几何分析是几个传统学科的混合领域：图论、几何、离散群理论和概率论。正如标题所示，这个领域只关注图的分析，图是“一维细胞复合体”的同义词。然而，作为一维对象的图中的边，除非我们讨论图的几何，否则不会发挥实质性的作用。因此，我们的观点与量子图的情况不同，例如，在量子图中，边缘上的微分算子是至关重要的。实际上，边在分析中的作用只是给出一个顶点之间的相邻关系，与此关系相联系的差分算子代替微分算子。因此，我们不必担心函数的不规则性，对于微分算子来说，不规则性可能会引起一些麻烦，如果不是很严重的话。相反，图形的组合方面创造了一种不同的技术复杂性。此外，对无限图和非紧流形的分析涉及的困难程度大致相同。我们研究的“主角”是流形上拉普拉斯算子（及相关算子）的离散模拟，它出现在数学科学的许多部分。在事物的本质上，全局分析中培养的思想一方面为我们提供了良好的指导原则，另一方面，实践动机是理论进步的推动力。离散拉普拉斯算子在几何晶体学和概率论中都有出现。在我们的研究中，我们应用晶格上随机行走理论的一个显著结果来确定金刚石晶体。将结果重新表述为“晶格上的随机步行者可能会探测到他的晶格在空间中最自然的方式”是很有趣的。金刚石孪生体是指具有金刚石晶体所满足的对称性质的假想晶体。我们的结果表明，作为完全图K_4的极大阿贝尔覆盖图的标准实现，只有菱形孪生。在活动方面，我是在剑桥大学牛顿研究所举办的特别项目的组织者之一，该项目从2007年1月开始，到2007年6月结束。少

Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. As indicated by the title, this field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". Edges in a graph as 1-dimensional objects, however, do not play a substantial role except when we discuss geometry of graphs. Therefore our view is different from, for instance, the case of quantum graphs where differential operators on edges are vital. Actually the role of edges in analysis is just to give a neighboring relation among vertices, and difference operators linked with this relation replace differential operators. We thus do not need to worry about irregularity of functions which, for differential operators, may cause some trouble, if not serious. Instead, the combinatorial aspect of graphs creates a different kind of technical complication. Furthermore, analysis on both infinite graphs and non-compact manifold … More s involves much the same degree of difficulty.The "protagonist" in our research is a discrete analogue of Laplacians on manifolds (and related operators), which appears in many parts of mathematical sciences. In the nature of things, ideas cultivated in global analysis provide us a good guiding principle on the one hand, and the practical motivation is the moving force of theoretical progress on the other..Discrete Laplacians appear in both geometric crystallography and probability.In our research, we applied a remarkable result in the theory of random walks on crystal lattices to pin down a diamond crystal. It is interesting to rephrase the result such as "A random walker on a crystal lattice may detect the most natural way for his crystal lattice to sit in space". A diamond twin means a hypothetical crystal which shares the property of symmetry satisfied by the diamond crystal. Our result claims that there is only diamond twin, which is obtained as the standard realization of the maximal abelian covering graph of the complete graph K_4.As for the activity, I was a member of organizers of the special project held at Newton Institute, Cambridge University which stated from January and ended up in June, 2007. Less